To refine search, see subtopics Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Five Fold Symmetry, Penrose Tilings, Double Frieze Patterns, Two Sided Frieze Patterns, Rotational Symmetry Groups (Rosettes), Bichromatic Plane Patterns, and Dynamic Symmetry. For more material on this topic, see subtopics Pattern and The Regular Solids. To expand search, see Art, Discrete Mathematics, and Geometry. For material on related topics, see Group Theory and Pattern. Laterally related topics: Perspective, Fractals in Art, Weaving, Renaissance Art, Basket Making, Tattoos, Pottery, Pattern, Architecture, Proportion and the Golden Ratio, Metal Work, Knots and Knotwork, Wood Carving, Bronzework, Needlework, Art History, Origami, Mazes, Graph Theory, Combinatorics, Tilings, Information Theory, Analytic Geometry, Trigonometry, Geometric Theorems, The Pyramid, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, Irrationals, The Pentagram, The Sphere, The Conic Sections, Polygons, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, and The Square.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Artmann, Benno. The cloisters of Hauterive. Math. Intelligencer 13 (1991), no. 2, 44--49. SC: 00A69 (01A99), MR: 1 098 219.
The author discusses geometric principles behind Gothic tracery. The Gothic style developed in France about 1150, but spread widely in the next few centuries. Examples are taken from Reims, Haina, Strasbourg, and Esslingen. The geometric principles are by no means trivial; some make rather challenging exercises. The author discusses the windows of the cloisters of Hauterive in some detail. Hauterive is a Cistercian monastery near Fribourg in Switzerland, and the cloister dates from 1320-1328. The windows there are unusually geometric, and the author advances the theory that the windows amount to a kind of commentary on Book IV of Euclid's Elements. One window, however, can not be constructed with straightedge and compass: it involves the construction of a regular 9-gon. The author notes that a regular 15-gon may have originally been envisioned, but that "esthetic considerations overwhelmed mathematics." Interesting article. A number of illustrations, a few of which appear in Artmann, Benno; Swetz, Frank J., The Geometry of Gothic Church Windows. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, Euclid, and Religion.
Artmann, Benno; Swetz, Frank J. The Geometry of Gothic Church Windows. In Swetz, Frank J. From Five Fingers to Infinity. A Journey through the History of Mathematics. Open Court, Chicago, 1994. 228.
Illustrations adapted from Artmann, Benno, The cloisters of Hauterive. The tracery in European Gothic churches uses arcs of a circle, fitted together in ingenious ways. Some of the ingenious ways have mathematical principles underlying them. Although this brief excerpt does not mention it, it is not uncommon for the construction to be repeated in the same tracery in a different scale---a kind of reaching to infinity that is reminiscent of fractals. Closely related topics: Medieval Europe, France in the Middle Ages, Fractals in Art, Similarity, Rotational Symmetry Groups (Rosettes), Polygons, The Circle, and Religion.
Bérczi, Sz. Symmetry and technology in ornamental art of old Hungarians and Avar-Onogurians from the archaeological finds of the Carpathian Basin, seventh to tenth century A.D. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 715--730. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058b.
Analysis of symmetries can be very helpful in better understanding archaeological art and artifacts. The types of symmetries not only show what the author describes as "intuitive mathematical development in ornamental art" but can also help trace relationships between different communities. Such studies are now relatively new, but with time should become "an accepted, standard part of the description of archaeological finds". In this article, the author discusses how all 7 types of strip/frieze patterns occur in Old Hungarian ornamental art, and develops a notion of a double frieze pattern, which is intermediary between frieze patterns and plane patterns. A number of these patterns occur (sometimes individualized) in Avar-Onogurian artifacts. The author's classification of double frieze patterns focuses on how the patterns are generated horizontally and vertically, and may be more useful for archaeological purposes than classification by the related plane patterns. The author gives examples of some plane patterns that came up somewhat naturally, including patterns from weaving, chained ring structures, and the optimal fitting of furs (a pmg plane pattern). The author compares the frequencies of certain symmetry patterns in collections from several cultures. Closely related topics: Hungary in the Middle Ages, Frieze Patterns, Plane Patterns, Double Frieze Patterns, Archaeology, and Metal Work.
Campbell, P. J. The geometry of decoration on prehistoric Pueblo pottery from Starkweather Ruin. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 731--749. (Reviewer: M. P. Closs.) SC: 01A12 (92A90), MR: 90h:01003.
Starts by introducing the mathematical principles behind classifications of symmetry groups for strip or frieze patterns and the plane patterns, and briefly discusses some other symmetry groups. Next, reviews the literature of the papers that have used symmetry patterns to classify and analyze designs. All an excellent introduction. The remainder of the article applies these methods to the later Pueblo pottery at Starkweather Ruin (Tularosa black-on-white and Reserve black-on-white). Ends with a discussion of to what extent the work of these and similar potters was mathematical. Closes with a quotation by Schattschneider on the work of "amateurs": "The mind and spirit are the forte of all such amateurs---the intense spirit of inquiry and the keen perception of all they encounter. No formal education provides these gifts. Mere lack of a mathematical degree separates these 'amateurs' from the 'professional'. Yet their dauntless curiosity and ingenious methods make them true mathematicians." Closely related topics: Archaeology, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, and The Pueblo Indians.
Chorbachi, W. K. In the tower of Babel: beyond symmetry in Islamic design. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 751--789. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A30 92K99), MR: 91a:01058c.
An interesting and personal account of how the author discovered geometric manuscripts written for Islamic artisans. With this discover, the author gives a new historical and scientific basis to the study of certain kinds of Islamic art. Much work preceding the author's had focused on religious, mystical, or perceptual interpretations of the work. Many ideas were primarily hypothetical, such as the (incorrect) idea that all Islamic art derives from the circle. The author suggests that many religious and mystical interpretations of Islamic geometric art should not be regarded as being historically based. Instead, the author shows how some Islamic art is highly mathematical, showing concerns with such topics as Pythagorean triangles and the notion of similarity (he gives an example where a shape appears in three different scales, each similar shape being derived from the last by a clever process). Much of the article discusses these in the context of a cyclic quadrilateral appearing in Islamic art with sides 1, 2, 2, 71/2. The author even noted an Islamic anticipation of a shape used to produce Penrose tilings. The author suggests that symmetry groups, while useful, can not alone give a full understanding of Islamic art. Closely related topics: The Islamic World, Art, Plane Patterns, Pythagorean Triangles and Triples, Penrose Tilings, Religion, and Mathematics and Mysticism.
Cromwell, Peter R. Celtic knotwork: mathematical art. Math. Intelligencer 15 (1993), no. 1, 36--47. SC: 01A07 (00A69), MR: 1 199 275.
Cromwell discusses a theory for the construction of Celtic knot friezes. These knot patterns may have been inspired by basketry (or maybe by textiles). He then analyzes the patterns in the knot friezes using a notion of a two-sided frieze pattern. There turn out to be 31 such patterns; 7 of these are the standard monochromatic strip patterns; 17 are exactly analogous to the bichromatic strip patterns; and 7 are like the monochromatic strip patterns but require the two sides to be identical. These last 7 "grey" patterns can't occur in knotwork, since the two sides of a crossing are not identical. Of the 24 monochromatic and bichromatic patterns, 12 cannot occur in Celtic knotwork because they would require strings that don't tie up, and 2 require a string straight through the centerline (and also don't occur). The other 10 can theoretically appear. Of these 10, two do not seem to occur at all, and one occurs but with an apparently different constriction technique (an example of this type is thought to be Scandinavian). The author is able to explain the rareness of these symmetry types in terms of the theory for their construction and from the fact that Celtic know friezes were generally finite and had their ends knotted together; these constraints require construction with an even grid, and the three problematic patterns require construction with an odd grid. This explains the type which does occur appears to use a different construction technique. In fact, the author found only one Celtic pattern that uses an odd grid. (And of course it can't be used in a bounded way, though it can be used in a kind of border.) All 7 of the monochromatic frieze patterns were apparently used in generating the existing know patterns, assuming the theory of construction is true (the author makes no claims that it is). The author includes examples of his own for the 3 problematic odd-grid know patterns. Excellent article. The author includes a good bibliography of related topics. It goes as far as Norwegian peasant art, for example. Not inordinately technical, in spite of the way it might sound. Closely related topics: The Celts, Knots and Knotwork, Two Sided Frieze Patterns, Frieze Patterns, Bichromatic Strip Patterns, Weaving, and Basket Making.
Crowe, D. W. and Washburn, D. K. Groups and geometry in the ceramic art of San Ildefonso. Proceedings of the conference on groups and geometry, Part A (Madison, Wis., 1985). Algebras Groups Geom. 2 (1985), no. 3, 263--277. (Reviewer: H. S. M. Coxeter.) SC: 05B45 (00A05 01A12 20F32 52A45), MR: 87k:05055.
Discusses the types of frieze patterns and bichromatic strip patterns occurring in the pottery of the pueblo of San Ildefonso in New Mexico. The people of San Ildefonso are Tewa speaking and are thought to be of Anasazi descent. However, it should be noted that the pottery has apparently been influenced by the Spanish and by attempts to make it more readily salable. All 7 of the strip patterns and 14 of the 17 possible bichromatic strip patterns are exhibited. (The authors supply the missing 3 bichromatic strip patterns in a similar style. The authors supplement their discussion with an explanation of the appealing Coxeter notation for classifying the bichromatic patterns (the standard classification system is cumbersome) and give a table of the correspondences between various systems. A historical aside briefly discusses the study of plane patterns in the context of the Alhambra, where there is still some disagreement on which patterns are represented. Closely related topics: The Pueblo of San Ildefonso, Frieze Patterns, Bichromatic Strip Patterns, Plane Patterns, Pottery, Archaeology, The Islamic World, and Spain in the Middle Ages.
Crowe, Donald W. Erratum to: "The geometry of African art. I, II" (J. Geometry 1 (1971), 169--182; Historia Math. 2 (1975), 253--271). Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Math. 2 (1975), no. 4, 617. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986c.
The articles Crowe, Donald W., The geometry of African art and Crowe, Donald W., The geometry of African art interchange the names of the symmetries p3m1 and p31m in several places. Closely related topic: Plane Patterns.
Crowe, Donald W. The geometry of African art. III. The smoking pipes of Begho. The geometric vein, pp. 177--189, Springer, New York-Berlin, 1981. (Reviewer: M. P. Closs.) SC: 01A10 (51M20), MR: 84b:01004.
Introduces the strip and plane patterns. Gives a useful flowchart for recognizing them (and some examples). Then classifies the patterns appearing in smoking pipes from the Krama quarter of Begho, in Ghana. The most common strip pattern is the one usually referred to as pmm2 (number 7 in the author's own system). The most common plane patterns are pmm and p4m. As the author notes, both of these can be easily created as rows of pmm2 strips. Representatives of all 7 strip patterns were found, but only 7 of the 17 possible plane patterns occurred. The author also considered questions on the relative preponderance of the various strip types by four different levels in the dig; no noticeable differences were found. Closely related topics: Ghana, Frieze Patterns, Plane Patterns, and Archaeology.
Crowe, Donald W. The geometry of African art. II. A catalog of Benin patterns. Historia Math. 2 (1975), 253--271. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986b.
Discusses the strip patterns and plane patterns occurring in Benin art. All 7 strip patterns and 12 of the 17 frieze patterns occur, though about five of the frieze patterns which do occur are rare: two may only occur once, and one of these may be based on a European model. The author compares the Benin patterns with the Bakuba patterns. Glide reflections are more rare in Benin art than in Bakuba art, possibly because glide reflection symmetries may arise most naturally from weaving patterns. Benin art also tents to be more representational, Bakuba art more abstract. The author also considers Benin patterns to be less varied than Bakuba patterns. However, it appears that the bronzework itself is nearly unsurpassed. A catalog is given with most of the strip patterns the author has found in Benin art, along with one example of each of the 12 plan patterns that occur. The author does not discuss this, but some patterns combine elements of different symmetries: the authors example of a p1 symmetry would have been classified differently if either of its two motifs were removed. Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art. Closely related topics: BeninCity, Nigeria, Frieze Patterns, Plane Patterns, The Bakuba of Zaire, Weaving, and Bronzework.
Crowe, Donald W. The geometry of African art. I. Bakuba art. J. Geometry 1 (1971), 169--182. (Reviewer: M. P. Closs.) SC: 01A15 (20H15), MR: 58 #9986a.
Discusses strip and plane patterns occurring in Bakuba art, particularly in textiles and woodcarving. The inspiration for many of these patterns seems to be from weaving, but at least one pattern may originate in the technique of sewing together triangles to make bark cloth. All seven strip patterns occur, and 12 of the 17 possible plane patterns. Discusses the relative proportions of some of these patterns, and gives an example of each. In all but one of the strip patterns, the author gives both cloth and carved examples (the other is given in cloth only, being rare in wood). The author includes an appealing claim about one of the patterns, made by an earlier researcher (too enthusiastic in the view of the authors): "it is probably the most remarkable example of this kind... its discovery is certainly a mathematical accomplishment of the first magnitude." Also see the erratum, Crowe, Donald W., Erratum to: "The geometry of African art. Closely related topics: The Bakuba of Zaire, Frieze Patterns, Plane Patterns, Weaving, and Wood Carving.
Doczi, György. Seen and unseen symmetries: a picture essay. Symmetry: unifying human understanding, I. Comput. Math. Appl. Part B 12 (1986), no. 1-2, 39--62. SC: 92A27 (01A99 52-01), MR: 838 136.
Certainly an unorthodox essay. It may be hard to understand the author's terms dinegy and dinergic symmetry (involving the union of complementary opposites), at least in a concrete mathematical sense, but the discussion and pictures do emphasize how mathematical proportions can pervade both art and the natural world. Closely related topic: Proportion and the Golden Ratio.
Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.
Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Vitruvius, Architecture, Proportion and the Golden Ratio, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
Gerdes, P. Reconstruction and extension of lost symmetries: examples from the Tamil of South India. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 791--813. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 92K99), MR: 91a:01058d.
Gerdes discusses the designs drawn (or formerly drawn) by Tamil women in South India during the harvest month Margali. The author shows that some of the diagrams may be degradations of earlier patterns that display more symmetry and/or are constructed according to the cultural ideal of having only one line. Gerdes also discusses drawing algorithms; many algorithms work by applying a series of simple transformation rules to a simpler motif. The function of these diagrams appears to be religious. As the author explains, "Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali." Closely related topics: The Tamil of South India, Continuous Tracing Problems, and Religion.
Gerdes, Paulus. Fivefold symmetry and (basket) weaving in various cultures. Fivefold symmetry, 245--261, World Sci. Publishing, River Edge, NJ, 1992. SC: 52B99 (01A07), MR: 1 178 750.
Gerdes suggests that five-fold symmetries arose from efforts to solve problems in basketweaving rather than in observations of five-fold symmetry in natural phenomena (such as starfish). One way five-fold symmetries can arise is by modifying the more obvious six-fold symmetries (such as those used by peasants in Mozambique) to fit a curved surface. The author reports that "these pentagonal-hexagonal baskets are, for instance, also woven by the Ticuna and Omagua Indians (northeastern Brazil), by the Huarani Indians, by the Kha-ko in Laos, and by the Menda in India. One sees them also in China, Japan, and Indonesia." The Malaysian sepak tackraw ball is similar to the soccer ball and is woven in the same way. The author reports that the peasants of the island Roti (Indonesia) may have discovered a way to fold a regular pentagon as a kind of a thimble. The author shows how a similar pentagonal weaving pattern is used in weaving brooms in Mozambique. (A near pentagram then appears inside the knot.) The author notes that a similar method is used in Angola to hold together the bars of a cage. The author in addition discusses how hat weaving techniques can lead naturally to three- and five-fold symmetries. The author's main example is with the hats of the Belu of central Timor, but he notes that related techniques are used in northern Mozambique, southern Tanzania, and by the Kuva of Congo. The author also shows a Chinese hat with five-fold symmetry. Two other particularly interesting examples are "a burden basket ... from the Papago Indians (Arizona) which combines beautifully a global sevenfold symmetry with local fivefold symmetry", and the "center of a Japanese basket, which combines global ninefold symmetry with local fivefold symmetry." Closely related topics: Five Fold Symmetry, Basket Making, Mozambique, Malaysia, and The Belu of Central Timor.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).
Gerdes, Paulus P. J. On ethnomathematical research and symmetry. Symmetry in a kaleidoscope, 2. Symmetry Cult. Sci. 1 (1990), no. 2, 154--170. SC: 01A07, MR: 1 188 949.
Gerdes begins with a discussion of why symmetry is such a common phenomenon in human culture. He notes that some symmetries which are rare in nature (e.g., rotational symmetries of order 2) are common amongst us. Gerdes gives the example of rotational symmetry being used in the tattoos of the Makonde of northern Mozambique. Gerdes explains how symmetries such as the rotational symmetry of order 2 can arise naturally in solving problems in such areas as weaving. Gerdes then turns to the geometry of the line drawings made by the Tamil women in South India (during harvest month) and those made by the Tshokwe. These drawings have some strong similarities, and in both cases show an interest in tracing out a figure with a single continuous line. They also show a strong interest in symmetry, and Gerdes gives examples of how designs which fail to follow the one-line cultural norm may also fail to display the expected symmetries, suggesting that such drawings are degradations of more symmetric ones drawn with one line. The author advances a construction principle that can be used to construct both the Tamil and Tshokwe patterns. (Although the author doesn't note this, it is interesting that this principle is very similar to another principle that has been advanced for Celtic knot friezes!) Gerdes then discusses some mathematical properties of curves made using his construction principle. He also discusses some other interesting topics in his ethnomathematical research. For example, the author mentions that he has a found a new hypothesis on the origin of the Egyptian formula for the volume of a truncated pyramid, and has also found an infinite series proof for the Pythagorean theorem. Closely related topics: The Tamil of South India, TheTshokwe, Continuous Tracing Problems, The Celts, Ancient Egypt, and Pythagorean Triangles and Triples. Also possibly relevant: Mozambique, Tattoos, and Weaving.
Gerdes, Paulus and Bulafo, Gildo. Sipatsi. Technology, art and geometry in Inhambane. Translated from the Portuguese by Arthur B. Powell and Gerdes. Instituto Superior Pedagógico, Ethnomathematics Research Project, Maputo, 1994. 102 pp. (Reviewer: J. S. Joel.) SC: 01A07 (00A08 00A69 01A13 51M20), MR: 95f:01002.
The authors discuss the construction and mathematical properties of the Mozambican sipatsi, which are essentially woven handbags. They are generally decorated with strip or frieze patterns, and in fact all 7 possible types of strip patterns occur in the sipatsi from Inhambane province in Mozambique. This book includes a description of the processes used to create the sipatsi, a catalog of the strip patterns found, and a chapter designed for people using the sipatsi to teach mathematics. The authors also give just a few examples of strip patterns on wooden spoons (also from Inhambane province) and on vases and pots (from Maputo). Closely related topics: Mozambique, Basket Making, Frieze Patterns, and Education.
Grünbaum, Branko. The emperor's new clothes: full regalia, G-string, or nothing? With comments by Peter Hilton and Jean Pedersen. Math. Intelligencer 6 (1984), no. 4, 47--56. (Reviewer: H. S. M. Coxeter.) SC: 01A15 (01A60 05B45 20F32 52A45), MR: 86d:01004.
Grünbaum's article: The author discusses the common misconceptions that the Egyptians and the artists of the Alhambra had used all 17 types of plane patterns. In fact, the Egyptians appear to have missed the five symmetry groups which have three-fold rotations. The sources for these misconceptions are discussed as well. The author has done fairly extensive research on the subject, and has concluded that two of the four plane patterns missing from the Alhambra seem not to appear at all in Islamic art (these are pg and pgg; the two missing at the Alhambra but present elsewhere are p2 and p3m1). A final theme of the author's is that the language of symmetry groups may at times be inadequate to discuss patterns, and can also be misleading in connection with the intentions of the artists themselves.The response by Peter Hilton and Jean Pedersen: The author's acknowledge Grünbaum's correction about the Egyptians. The authors note that the Egyptians and Moore's between them only missed one symmetry group, p3m1. They comment briefly on Chinese and Japanese designs, and quote Schattschneider, who notes that Chinese and Japanese artwork features rotations and glide reflections much more strongly than Islamic art does. Schattschneider also cites an illustration from a Japanese book that seems to suggest that underlying lattices of squares, equilateral triangles, rhombuses, and parallelograms were consciously used in developing symmetry patterns. The authors acknowledge the limitations of group theory in discussing symmetry, but also emphasize its usefulness. Closely related topics: Plane Patterns, Ancient Egypt, The Islamic World, Penrose Tilings, Japan, and China.
Groemer, H. The symmetries of frieze ornaments in Maya architecture. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 203 (1994), 101--116 (1995). (Reviewer: J. S. Joel.) SC: 01A12 (00A69 01A07 51M20 52C20), MR: 96b:01006.
The author discusses the frieze patterns that occur in Mayan architecture, with occasional references to the frieze patterns found on Mayan pottery. All seven basic types of frieze patterns occur, though one (the one with only glide reflections) is rather rare. The author notes that many of the symmetries appear to be derived from the symmetries of the same base motif, which is merely translated; it is acknowledged that this distinction is not a mathematical one. The author also distinguishes between discrete and continuous patterns. One interesting pattern is classified as having only translations and vertical reflections, but as the author notes, the "negative space" has an upside-down version of the same ornament This particular pattern could be classified as a bichromatic strip pattern, but apparently the "negative space" symmetry is lost in some other examples of the motif. The author finds, to his surprise, that there seems to be little Toltec or Zapotek-Mixtec influence in the Mayan frieze patterns. Closely related topics: The Maya, Architecture, Frieze Patterns, and Pottery.
Grünbaum, Branko, Grünbaum, Zdenka; Shephard, G. C. Symmetry in Moorish and other ornaments. Computers \& Mathematics with Applications. Part B 12 (1986), no. 3--4, 641--653.
It is observed that 13 of the 17 plane patterns are represented at the Alhambra. Two of the four missing groups have been found in Toledo, Spain, and dating from about the same period (one, p3, was found in a church, and the other, p3m1, was found in a synagogue). The authors note that the remaining two patterns (pg and pgg) seem not to appear in Islamic art at all. The authors note that features of Islamic art are not always fully described by the symmetry groups alone; such features can include color changes and interlace patterns. The color-symmetry groups are only a partial solution to the former, since colors are often in ratios "2:1:1, 4:2:1:1, 6:2:1, 6:3:1:1:1 or some similar ratio... The mathematical theory of such colorings still awaits development." The authors also attack the commonly held view that the artists of the Alhambra exhausted the possibilities of symmetry in art, and illustrate their points with pictures. Moreover, the authors suggest that ideas of local structure are as important as ideas of global structure. "The various kinds of symmetry groups are useful in the description of many of the artifacts, but more general approaches (based on 'adjacency relations' or other 'local' criteria) are necessary for a better understanding of the ornaments and artwork, and of the ways their creator thought about them." Closely related topics: The Islamic World and Plane Patterns.
Grünbaum, Branko; Shepard G.C. The geometry of fabrics. Geometrical Combinatorics, 77--97, Pitman, Boston, 1984.
Symmetry groups have already been used in discussing mathematical properties of textiles, but may not be appropriate for all kinds of fabrics. In this article, the author discusses primarily isonemal fabrics. In these, the symmetry group acts transitively on the strands of the fabric. Many fabrics that actually occur in nonmathematical discussions are actually isonemal. This article (and for example its classification of isonemal fabrics into 5 genera) could form part of the foundation for a new ehtnomathematical research area. In addition, a number of interesting combinatorial questions arise. The article focuses primarily on traditional fabrics, with two perpendicular layers. The terminology of the article may be less appropriate for the three layer fabrics that sometimes occur in basketweaving. "It is interesting to note that in the case of 3-way 3-fold fabrics some 'partial fabrics', that is parts of fabrics that do not hang together, are used in basketry... The classification of such fabrics seems to be a totally unexplored area of the subject." With regard to 3-way fabrics, he notes that "it seems that such fabrics are more stable under diagonal strain than 2-way 2-fold fabrics, and so have been used in such practical applications as parachutes." Similar concerns may of course account for their occurrence in basket making as well! Closely related topics: Weaving, Combinatorics, and Basket Making.
Grünbaum, Branko; Shephard G. C. Interlace patterns in Islamic and Moorish art. The Visual Mind, 147--155, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
Many Islamic and Moorish patterns exhibit what the authors call interlace patterns, where the patterns seem to be made of strands that alternately go over and other strands. This is a phenomenon that makes these Islamic artworks appear something like a 2-D extension of the Celtic knot friezes; the over/under rule is of course also common in weaving. The authors focus on the seemingly curious phenomenon that many of the Moorish and Islamic interlace patterns can be viewed as being made of a small number of basic shapes, often one or two. The authors analyze this phenomenon for the symmetry groups p4m and p6m, and find that it arises in a mathematically natural way, especially if artists used stencils, as is sometimes now thought. The article gives give propositions without proof; proofs of these should be within reach of a good undergraduate with the requisite knowledge of group theory. Closely related topics: The Islamic World, Plane Patterns, Knots and Knotwork, and Weaving.
Hargittal, István and Lengyel Györgi. The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Journal of Chemical Education 61 (1984), 1033.
All 7 frieze patterns can be found in Hungarian needlework. The authors give an example of each pattern. A related article by the same authors is Hargittal, István and Lengyel Györgi, The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary, and Needlework.
Hargittal, István and Lengyel Györgi. The seventeen two-dimensional space-group symmetries in Hungarian folk needlework. Journal of Chemical Education 62 (1985), 35--36.
The Hungarians of the late 1800s may be among the earliest people known to have "discovered" all 17plane patterns. The authors give an example of each pattern from Hungarian needlework. For the related article on frieze patterns, see Hargittal, István and Lengyel Györgi, The seven one-dimensional space-group symmetries illustrated by Hungarian folk needlework. Closely related topics: Frieze Patterns, Hungary in the 1800s, and Needlework.
Jablan, Slavik. Geometry in the pre-scientific period. Geometry in the pre-scientific period; ornament today, 1--32, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 91i:01004.
Discusses geometric ornamentation in Paleolithic and neolithic mathematics, focusing on the symmetries in the ornamentation. The author gives many examples. The only possible symmetry groups of the rosettes are Cn and Dn. There are infinitely many of these, of course, but the basic types occur in both the Paleolithic and the Neolithic. There is a somewhat wider variety in the Neolithic. In addition, neolithic artists have also explored some of the corresponding antisymmetry (or bichromatic) groups. It turns out that all 7 of the frieze already occur in the art of the Paleolithic; thus not surprisingly they occur in the art of the Neolithic as well. The examples show that there are interesting differences in the ways that the frieze patterns are applied. 14 of the 17 bichromatic strip patterns (antisymmetry groups) occur in neolithic ornamental art. 14 of the 17 plane patterns occur in the Neolithic. The author discusses reasons why the artists may have explored the patterns that they did. The author also finds 23 of the bichromatic plane patterns, and gives an example of each. (He classifies these using the Coxeter group/subgroup notation.) Closely related topics: The Paleolithic Era, The Neolithic Era, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, and Rotational Symmetry Groups (Rosettes).
Jablan, Slavik. Ornament today. Geometry in the pre-scientific period; ornament today, 33--65, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 92g:01008.
The author discusses how a wide variety of mathematical notions can be used to help describe and understand the patterns occurring in art. One of the most important is, of course, the notion of symmetry, including those in the rotational symmetry patterns, frieze patterns, plane patterns, and their bichromatic (or antisymmetry) variants. More complex types of patterns also occur in art, and as Grünbaum, Grünbaum, and Shephard observed in their article Symmetry in Moorish and other ornaments, many of the problems originating from these are still unsolved. Examples are given from the Paleolithic to the 20th century. The author touches on (to give a few examples) interlace patterns (often considered to be connected with weaving), similarity symmetry, symmetries in higher dimensional spaces, and on some of the ideas of the theory of tilings, including Penrose tilings and hyperbolic tilings. The author also gives examples from the work of artists including M. C. Escher, B. Riley, and R. Neal. A fine article. A fine article. It could easily take a class an entire semester to examine in detail all the ideas presented. Closely related topics: Art, Pattern, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Penrose Tilings, Weaving, Similarity, and M. C. Escher.
Knight, Gordon. The geometry of Maori art---Rafter Patterns. New Zealand Math. Mag. 21 (1984), no. 2, 36--40.
The Maori have been fond of carving patterns on their rafters. The author wondered if all seven possible strip or frieze patterns occur in the work of the early Maori, and he found in fact that they do. There is also a brief discussion of the seven types of strip patterns and a flowchart for recognizing them. The author's source was the book Maori Art by A. Hamilton (N.Z. Institute, Wellington, 1901), which is now reprinted by Holland Press, London, 1972. Hamilton's book illustrates 29 rafter patterns, and these turned out to have had only six of the seven patterns; fortunately the one with only vertical reflections turned out in a photograph elsewhere in the book, in "part of the porch of a large house of the Ngati-Porou at Wai-o-Matatini". The author lists two questions that he does not have answers to: What was the relative frequency of each group, and did this vary from one tribal region to another? Also, is there a geometrical difference in character between the early Maori patterns and those produced after the influence of the Pakeha? Closely related topics: The Maori, Wood Carving, and Frieze Patterns.
Knight, Gordon. The geometry of Maori art---weaving patterns. New Zealand Math. Mag. 21 (1984), no. 3, 80--86. (Reviewer: H. S. M. Coxeter.) SC: 51N20 (01A10), MR: 87m:51059.
If one restricts only to 90 degree weaving, only 12 of the 17 plane patterns are possible as symmetry groups. 10 of these 12 plane patterns are represented in Maori art. The article gives an example of each. There is also a simple flowchart for recognizing the 17 symmetry groups of the plane patterns. As an additional aid in recognition, the author also includes a couple of examples of plane patterns which he labels with possible translation vectors, points of rotation, and lines that can be used in reflections and glide reflections. The author does not discuss whether weaving of the 120 degree type occurs in Maori art. Closely related topics: The Maori, Weaving, and Plane Patterns.
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Proportion and the Golden Ratio, Leonardo of Pisa (Fibonacci), The Pentagram, and Education.
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Philosophy, Greece, Physics, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
Mamedov, Kh. S. Crystallographic patterns. Symmetry: unifying human understanding, II. Comput. Math. Appl. Part B 12 (1986), no. 3-4, 511--529. SC: 00A69 (01A99 20H15 51F15), MR: 87e:00008.
This article discusses how crystallographic patterns "and their distribution and connection with natural phenomena and subjects of pure and applied art." It is written as an essay from a personal point of view. As the author tells us "I have made no effort to restrict the style of my meditations. I have presented a flow of free and sincere statements, and have not attempted to impose on them a style which might conceal their individuality. A great advantage of such statements is that one's 'falsehoods' are merely considered to be delusions, thus somehow mollifying the anger of those strict critics who feel obliged to adhere to absolute truths." The author himself is a chemist, so it is not surprising that there is some discussion of how crystallographic patterns in art are similar to those in chemistry. However, his observations on art from his own background in a nomadic family from Azerbaijan may be at least as valuable. The author notes that M. C. Escher is often identified with the applied art of crystallographic patterns, but these ideas are common in many cultures. Crystallographic patterns involving elements such as colored symmetry "are very characteristic of ancient and medieval decorations of Siberia, Kazakhstan, Central Asia, Azerbaijan, and Asia Minor." Quite a few examples of the art in this article use Islamic khufic script, and as he notes it is common to attribute the rise of patterned art rather than representational art to religious demands. The author does not seem entirely sympathetic with this idea, writing "The the problem was 'explained with God's help.' It is evident that in such cases it is much easier for the representatives of some other tradition to invent a new explaining theory than to examine the artwork using the language of its own traditions." The author gives some examples of crystallographic patterns in his own art and that of his associates. Interesting and enjoyable article. Closely related topics: Plane Patterns, Religion, Language and Literature, and M. C. Escher.
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.
Discusses the history and philosophy of origami and then (in a little more depth) discusses some of its applications. The author discusses applications in math and science education, and also in art, design, and technology. A particularly interesting application of paper-folding and the theory of polyhedra is in music education, where one researcher devised "a 'tower' of five octahedra, to illustrate some basic concepts in musicology. His inspiration was from a work by Möbius written in 1861. Ganter's compound polyhedron illustrates geometrically the following concepts and their connections: the vertices correspond to the notes of the chromatic scale, the edges corresponds to the thirds and fifths, and the triangular faces correspond to the triads." He mentions that M. C. Escher was interesting in construction paper models (though it is not really clear how deep that interest lay). It is interesting that the well-known book by T. Sundara Row entitled Geometric Exercises in Paper Folding seems to be independent from the Japanese traditions. Closely related topics: Origami, Japan, Education, Music, M. C. Escher, and August Ferdinand Möbius (1790-1868).
Nagy, Dénes. The 2,500-year old term symmetry in science and art and its "missing link" between the antiquity and the modern age. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 18--28. SC: 01A99, MR: 1 371 622.
Documents the evolution of the word symmetry from its beginnings in ancient Greece. As the author explains, the word originally had a somewhat different meaning: symmetry = syn together + metron measure, suggesting the notion of commensurability. The word was adopted into Latin but was apparently rare in the middle ages. It's reappearance can probably be credited to the importance to the Renaissance of the De architectura libri decem of Vitruvius (1st century BC). The author discusses the Hebrew, Indian, and Chinese words for symmetry as well. At the end of the article the author enumerates some modern generalizations and uses of symmetry. For example, the author mentions "Noether's theorems connecting symmetry transformations (invariances) and conservation laws", Gell-Mann and Ne'eman's classification of elementary particles, and "Graeser's reconstruction of Bach's Kunst der Fuge". Closely related topics: Language and Literature, Greece, Vitruvius, Physics, and Music.
Schattschneider, Doris. The fascination of tiling. The Visual Mind, 157--164, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
As the author notes, "interlocking shapes displayed in majolica tile, inlaid wood, brickwork, carved stucco, stone pavement, sewn patchwork or printed fabric hold a special fascination for many people that goes far beyond the aesthetic pleasure that these patterns provide." The author gives the example of the artist M. C. Escher, who "often described regular divisions of the plane as the 'richest source of inspiration I have ever struck.'" This article is an excellent introduction to some of the mathematical problems that arise in the study of tilings. The author discusses for example uniqueness of tilings, Penrose tilings, Conway Criterion tilings, rep-tiles, and some of the issues that arise in the classification of tilings. She uses the example of rep-tiles to give a hint on how one can prove that certain tilings are aperiodic. She uses an artwork of M. C. Escher to illustrate three methods of classifying tilings. She mentions some other issues in tiling as well, such as tilings of non-planar surfaces; the references should help track these issues down further. Very clearly written. One minor comment is that a couple of times in the article she tells us that there is "no test or algorithm" that will answer a certain kind of question; it seems that she may actually mean only that there is no test or algorithm currently known. Closely related topics: Tilings, Penrose Tilings, M. C. Escher, and Art.
Schattschneider, Doris. The plane symmetry groups: their recognition and notation. American Mathematical Monthly 86 (1978), 439--450.
Discusses in detail the classification of plane patterns. Although the author avoids group-theoretic notation, she manages to bring out the group theoretic nature of the plane pattern groups more clearly than most other authors discussing these patterns. There is a very useful chart on the seventeen plane patterns that clearly labels the locations of the centers of rotation (with labels that distinguish the 2, 3, 4, and 6-fold centers), the axes of reflection, and the axes of glide-reflection. The chart may give a better understanding of the differences between the different symmetry groups than the flowcharts that appear in some other sources. The author discusses the generating regions for each of the plane patterns, and gives examples for each symmetry group of two set of generators of the group (except in the case of the pattern p1, where there is only one natural set of generators. She illustrates the plane patterns with lattices, most of which are from China. There are a couple of examples from the artwork of M. C. Escher as well. There is also a table cross-referencing notations used by different sources. There are six different notations in all; as the author notes, one the differences results from the common confusion between the groups p3m1 and p31m. Closely related topics: Plane Patterns, Group Theory, Art, M. C. Escher, and China.
Sizer, Walter S. Mathematical notions in preliterate societies. Math. Intelligencer 13 (1991), no. 4, 53--60. (Reviewer: U. D'Ambrosio.) SC: 01A07 (01A12 01A13), MR: 93a:01002.
The author discusses the ethnomathematics of nonliterate societies. There is little detail, as the article is rather brief, but the author does mention the number concept and counting, fractions (very briefly), elementary geometric notions (e.g., that of a line), symmetry, string figures, and games of strategy. One note on the article: there are strong similarities behind the mathematics in different parts of the world. There is a theory that this similarity is due to a common origin. The author credits Cantor for this idea. It was first fully developed, however, by Abraham Seidenberg. Closely related topics: Ethnomathematics General, The Number Concept, Fractions, Geometry, Games, and String Figures. Also possibly relevant: Abraham Seidenberg.
Toth, Nicholas. The prehistoric roots of a human concept of symmetry. Symmetry in a kaleidoscope, 3. Symmetry Cult. Sci. 1 (1990), no. 3, 257--281. (Reviewer: J. S. Joel.) SC: 01A10 (00A99), MR: 93g:01005.
The author discusses how concepts of symmetry occur in Paleolithic artifacts such as stone tools and "Venus" figurines, and also in the roughly circular areas such as those used in a hut or even perhaps at Olduvai site DK 1 (some million years ago). The author has also noted some asymmetries in the making of flaked stone tools. "This slight but statistically significant patterning of asymmetry and possible preferential right-handedness between 1.9 and 1.5 million years ago may indicate a more profound specialization (lateralization) of the left and right hemispheres of the hominid brain by the early stone age." Closely related topics: The Paleolithic Era, Archaeology, and Biology.
Zaslavsky, Claudia. Africa counts. Number and pattern in African culture. Prindle, Weber & Schmidt, Inc., Boston, Mass., 1973. x+328 pp. SC: 01A10, MR: 58 #20993.
This book is an excellent introduction to the mathematics of (primarily sub-Saharan) Africa. The best tribute to its importance may be in Gerdes, Paulus, On mathematics in the history of sub-Saharan Africa. Gerdes writes "In her classical study Africa Counts: Number and Pattern in African Culture ..., Claudia Zaslavsky presented an overview of the available literature on mathematics in the history of sub-Saharan Africa. She discussed written, spoken, and gesture counting, number symbolism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric forms, while Donald Crowe contributed a chapter on geometric symmetries in African art." Regarding geometric symmetries, it is primarily the frieze patterns and plane patterns that are discussed; there is surely more work to be done on the bichromatic frieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensive bibliography. Another useful resource is the newsletter distributed by the African Mathematical Union's Commission on the History of Mathematics in Africa (AMUCHMA). Closely related topics: Sub-Saharan Africa, TallySystems, Finger Numerals, Counting, Numerology, The Reckoning of Time, Money, Measurement, Games, Continuous Tracing Problems, Architecture, Magic Squares, Mathematics in Language, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.